An equilateral triangle is drawn in the $xy$ plane.

Prove that the coordinates of the vertices cannot all be rational numbers.

Without loss of generality, you can assume that one of the vertices is $(0,0).$

Let the other two be $(x,y)$ and $(a,b).$

Recall that each angle in an equilateral triangle is $\frac{\pi }{3}.$

Suppose, for contradiction, that $x,y,a,b$ are all rational.

Try to express $x$ and $y$ in terms of $\ell$ and $\theta .$

Then try to express $a$ and $b$ in terms of $x$ and $y.$

(Don't worry about calculating ${\ell }^2=a^2+b^2,$ it isn't needed for this particular way of proving.)

Why can't they all be rational?

This will help you get $a$ and $b$ in terms of $\ell$ and $\theta$ and therefore in terms of $x$ and $y.$